An Algebra Problem from RMO

Algebra Level 3

$\begin{cases} f(x)={ x }^{ 3 }+{ a }x^{ 2 }+bx+c \\ g(x)={ x }^{ 3 }+{ b }x^{ 2 }+cx+a \end{cases}$ Given that $a, b, c$ are integers with $c\neq 0$ and the following conditions hold for the above functions:

(a) $f(1) = 0$ ;

(b) the roots of $g(x) = 0$ are the squares of the roots of $f(x) = 0$ .

Find the absolute value of $({ a }^{ 2013 }+{ b }^{ 2013 }+{ c }^{ 2013 }).$